Available online at www.ispacs.com/cna Volume 2013, Year 2013 Article ID cna-00173, 5 Pages
doi:10.5899/2013/cna-00173 Research Article
Traveling wave solutions of a biological
reaction-convection-diffusion equation model by using
(
G
′
/
G
)
expansion method
Sh. Javadi1, M. Miri Karbasaki2, M. Jani1∗
(1)Department of Applied Mathematics, Kharazmi University, Tehran, Iran
(2)Department of Applied Mathematics, Islamic Azad University, Khash Branch, Khash, Iran
Copyright 2013 c⃝Sh. Javadi, M. Miri Karbasaki and M. Jani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, the(G′/G)-expansion method is applied to solve a biological reaction-convection-diffusion model
aris-ing in mathematical biology. Exact travelaris-ing wave solutions are obtained by this method. This scheme can be applied to a wide class of nonlinear partial differential equations.
Keywords:Expansion methods, Reaction-convection-diffusion equation, Nonlinear evolution equations, Exact Solutions
1 Introduction
Mathematical modeling of physical and biological systems often leads to nonlinear evolution equations. Exact solutions of these equations are of theoretical importance. Considerable efforts have been made to the study of solitary wave solutions. Various ansatze have been proposed for seeking traveling wave solutions of nonlinear differential equations. Recently, Many new methods have been proposed to find some particular solutions for these problems for instance, the Exp-function method [1, 2, 3], the homogeneous balance method [4, 5], the sine-cosine method [6], the rational expansion method and it’s generalizations [7, 8, 9], the Jacobi elliptic function method [10], the
F-expansion method and its extensions [11, 12, 13]. The objective of this paper is to use the(G′/G)-expansion
method to investigate exact traveling wave solutions of a biological model that is a reaction-convection-diffusion equation, namely Murray equation which can be considered as a generalization of the Fisher and Burgers equations.
The(G′/G)-expansion method is based on the explicit linearization of nonlinear evolution equations for traveling
waves with a certain substitution which leads to a second-order differential equation with constant coefficients [17-22]. Through the use of the method we can obtain more general solutions with some free parameters. Moreover, the
(G′/G)-expansion method transforms a nonlinear equation to a simple algebraic system of equations which can be
solved easily by means of a symbolic computational software like Maple, Matlab, Mathematica, etc. In this paper, we use Maple 15.
2 Description of the(G′/G)-expansion method
Wang et al. [15] summarized the main steps for using the(G′/G)-expansion method, as follows:
Step 1. The traveling wave variableξ=kx+wtpermits us to reduce the following partial differential equation
P(u,ut,ux,utt,uxt,uxx, . . .) =0, (2.1)
to an ordinary differential equation forU=U(ξ)of the form
Q(U,−wU′,kU′,w2U′′,k2U′′,kwU′′, . . .) =0. (2.2)
After solving the ODE (2.2), we have the some traveling wave solutionsu(x,t) =U(ξ)of the PDE (2.1).
Step 2. Suppose that the solution of the ODE (2.2) can be expressed as a polynomial in(G′/G)as follows:
U(ξ) = n
∑
i=0
αi(G′
G)
i, (2.3)
whereG=G(ξ)satisfies the second-order linear ODE in the form
G′′+λG′+µG=0, (2.4)
andαi,i=0, . . . ,n,λ andµare constants to be determined and the leading coefficientαnis nonzero. The
posi-tive integerncan be determined by considering the homogeneous balance between the highest order derivatives
and nonlinear terms appearing in (2.2).
Step 3. Substituting (2.3) into (2.2) and using (2.4), and then collecting all terms with the same power of(G′/G)
together, the left-hand side of Eq. (2.2) is converted into a polynomial in(G′/G). By equating each coefficient
of the resulted polynomial to zero, a system of algebraic equations is obtained forαi’s,k,w,λ,µthat can be
solved by a computational algebraic system (CAS) like Maple.
Step 4. Since the general solutions of the ODE (2.4) are well known for us, by substitutingαi’s,k,wand the general
solutions of Eq. (2.4) into Eq. (2.3), we have more traveling wave solutions of the nonlinear evolution equation (2.1).
Remark 2.1. The function G(ξ)which is the solution of Eq. (2.4) has the property that different order derivatives of the function(G′/G)can be expressed as a second order polynomial with respect to(G′/G). In fact, using (2.4) we have
d(GG′)
dξ =−(µ+λ(
G′ G) + (
G′ G)
2),
(2.5)
and so we have the following derivatives of U(ξ)
U′=dU
dξ =−(µ+λ(
G′ G) + (
G′ G)
2) dU
d(G′
G)
, (2.6)
U′′=d 2U
dξ2 = (µ+λ(
G′ G) + (
G′ G)
2)2 d2U
d(GG′)2+ (λ+2(
G′
G))(µ+λ,( G′
G) + ( G′ G)
2) dU
d(GG′), (2.7)
and etc. So by considering U(ξ)as Eq. (2.3), it follows that its derivatives are polynomials of(G′/G).
Remark 2.2.The general solution G(ξ)of Eq. (2.4) required in the last step has an explicit form and so the expression G′/G is as follows
G′(ξ)
G(ξ) =
√
λ2−4µ
2 (
C1sinh12
√
λ2−4µξ+C 2cosh12
√
λ2−4µξ
C1cosh12
√
λ2−4µξ+C 2sinh12
√
λ2−4µξ)−
λ
2, λ2−4µ>0,
√
4µ−λ2
2 (
−C1sin12
√
4µ−λ2ξ+C 2cos12
√
4µ−λ2ξ
C1cos12
√
4µ−λ2ξ+C 2sin12
√
4µ−λ2ξ )−
λ
2, λ2−4µ<0.
3 Application of the method
Consider the following reaction-convection-diffusion equation of the form
ut= (λ+λ0u)uxx+λ1uux+λ2u−λ3u2, (3.9)
whereλ,λ0,λ1,λ2andλ3are real constants [20]. In the particular caseλ =1 andλ0=0, this equation coincides
with the Murray equation
ut=uxx+λ1uux+λ2u−λ3u2, (3.10)
which itself is a generalization of the well-known Fisher equation whenλ1=0. When bothλ2andλ3are zero, it is
reduced to the classical Burgers equation. We introduce the traveling wave variableu(x,t) =U(ξ),ξ=kx+wtinto
Eq. (3.10) to find
wU′−k2U′′−λ1kUU′−λ2U+λ3U2=0, (3.11)
where prime denotes the derivatives with respect toξ. Considering the homogeneous balance between the highest
linear termU′′and the nonlinear termUU′in Eq. (3.11), the parameternthat is required in Eq. (2.3) is determined.
In fact by using (2.7), we haven+2=2n+1 and son=1. Therefore, the solution can be expressed as
U(ξ) =α0+α1(G′/G), (3.12)
whereα0andα1are constants to be determined later. Substituting Eq. (3.12) along with (2.6) and (2.7) into Eq. (3.11)
and collecting all terms with the same power of(G′/G)together, the left hand side of Eq. (3.11) is converted into a
polynomial in(G′/G). Equating each coefficient to be zero yields a set of simultaneous algebraic equations. Solving
this system by Maple, we get some trivial and nontrivial solutions. The trivial obtained solutions areU=0,λ2
λ3 and
the nontrivial traveling wave solution is as follows:
k=± λ1λ2 2λ3
√
λ2−4µ, w=±
λ2(λ2
1λ2+4λ32)
4λ2 3
√
λ2−4µ ,
α0=
λ2
2λ3
(1±√ λ
λ2−4µ), α1=±
λ2
λ3
√
λ2−4µ.
(3.13)
Case 1. Whenλ2
−4µ >0, by substituting (3.13) into Eq. (3.12) and using (2.8), the hyperbolic solutions are
obtained as follows:
U±
hyper(ξ) =
λ2
2λ3
(1±√ λ
λ2−4µ) ± λ2
λ3 √
λ2−4µ( √
λ2−4µ
2 (
C1sinh12 √
λ2−4µξ+C 2cosh12
√
λ2−4µξ
C1cosh12 √
λ2−4µξ+C 2sinh12
√
λ2−4µξ)−
λ
2),
(3.14)
whereξ= (± λ1λ2
2λ3√λ2−4µ
)x+ (±λ2(λ12λ2+4λ32)
4λ2 3
√
λ2−4µ )tandC1,C2are arbitrary constants. So we get different solutions
u(x,t) =U(ξ).
Case 2. Whenλ2
−4µ<0 by substituting (3.13) along with (2.8) into (3.12), and then simplifying the resulted
solution, we have the following trigonometric form
Utrig± (ξ) = λ2
2λ3
(1±√ λ
λ2−4µ)
± λ2
λ3
√
λ2−4µ(
√
4µ−λ2
2 (
−C1sin12
√
4µ−λ2ξ+C
2cos12
√
4µ−λ2ξ
C1cos12
√
4µ−λ2ξ+C
2sin12
√
4µ−λ2ξ )−
λ 2),
(3.15)
where the traveling wave is as
ξ =± λ1λ2
2λ3
√
λ2−4µx±
λ2(λ2
1λ2+4λ32)
4λ2 3
√
λ2−4µ t,
Remark 3.1. All of the derived solutions are tested by direct substitution in the studied problems to ensure the cor-rectness.
4 Conclusions
Traveling wave solutions to nonlinear evolution equations arising in mathematical biological systems are of theo-retical importance. Various ansatze have been proposed for seeking traveling wave solutions of nonlinear differential
equations. In this paper, the(G′/G)expansion method is successfully applied for obtaining exact traveling wave
solutions of a reaction-convection-diffusion equation (i.e., Murray’s biological model). It is shown that the(G′/G)
-expansion method is quiet efficient and well suited for finding exact solutions. The reliability of the method and reduction in the size of computational domain give this method a wider applicability. With the aid of Maple and by putting them back into the original equation, we have assured the correctness of the obtained solutions. Finally, it is worthwhile to mention that the method is straightforward and concise and it can be applied to other nonlinear evolution equations in engineering and the physical sciences.
Acknowledgements
The authors would like to thank the referee for his/her valuable comments.
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